Embedded, or immersed, approaches aim to minimize the computational costs associated with generating body-fitted meshes by employing fixed, often Cartesian, meshes. However, this boundary treatment introduces a geometric error of the order of the mesh size, which, if not properly addressed, can compromise the global accuracy of a high-order discretization. High-order embedded methods are used to appropriately correct the boundary conditions imposed on an unfitted boundary, thereby compensating for the aforementioned geometric error and achieving high-order accuracy. In this seminar, a stability analysis of discontinuous Galerkin methods coupled with embedded methods is conducted for the linear advection equation through the eigenspectrum visualization of the high-order discretized operators. Numerical experiments are presented to validate the stability analysis.